Linear theory of the spatial signatures of critical slowing down

Critical slowing down is considered one of the main signs that a dynamical system is close to a tipping point. For this reason, early-warning indicators have been developed to identify it, mainly based on the temporal evolution of the system under analysis. Yet, it has been shown that critical slowing down also displays spatial signatures, such as an increase in the spatial correlation. However, these signatures are based only on heuristic observations and the analysis of numerical simulations and do not have a sound theoretical foundation. In this work I will derive analytical expressions for typical spatial early-warning indicators of critical transitions in spatially extended systems, such as spatial correlation, spatial variance, and spatial permutation entropy, in the linearized limit. As a result, I will show that the common belief that the spatial correlation increases when a bifurcation is imminent is false. Other indicators, instead, might represent more robust alternatives, such as the spatial permutation entropy or the spatial variance.